Forgotten Fundamentals of the Energy Crisis - appendix

by Prof. Al Bartlett

Acknowledgements

A great deal of correspondence and hundreds of conversations with dozens of people over six years have yielded many ideas, suggestions, and facts which I have incorporated here. I offer my sincere thanks to all who have helped.

Appendix

When a quantity such as the rate r( t ) of consumption of a resource grows a fixed percent per year, the growth is exponential:

r ( t ) = r₀ e ^{k t} = r₀ 2 ^{t / T2}         (1)

where r₀ is the current rate of consumption at t = 0, e is the base of natural logarithms, k is the fractional growth per year, and t is the time in years. The growing quantity will increase to twice its initial size in the doubling time T₂ where:

T₂ (yr) = (ln 2) / k » 70 / P         (2)

and where P, the percent growth per year, is 100k. The total consumption of a resource between the present (t = 0) and a future time T is:

C = {T to 0} r(t) dt         (3)

The consumption in a steady period of growth is:

C = r₀ {T to 0} e ^kt dt = ( r₀ / k ) ( e ^kt - 1 )        (4)

If the known size of the resource is R tons, then we can determine the exponential expiration time (EET) by finding the time T_e at which the total consumption C is equal to R:

R = ( r₀ / k ) ( e ^kTe - 1 )         (5)

We may solve this for the exponential expiration time T_e where:

EET = T_e = ( 1 / k ) ln ( k R / r₀ + 1 )         (6)

This equation is valid for all positive values of k and for those negative values of k for which the argument of the logarithm is positive.